EU0011

euler_unit_product_factor_coprime

Public research checkpoint, not admitted to Alpha or Stable. Alpha v30 remains 3222 checked-use theorems; Stable remains 432. The on-demand Alpha Lean service does not yet expose these checkpoint theorems; their independently checked literal bundles and unchanged sources are available below.

Every weighted factor is coprime to the modulus, including the modulus-one zero factor.

Public research checkpoint: original HA and independently compiled Lean verified; not Alpha-enrolled, no Alpha checked-use authority; not Stable

The exact G014 theorem is proved in this research checkpoint for m>1 and genuinely invertible a. Phi counts coprime residues independently of the conclusion. The broader coprime theorem also handles m=1 by congruence, not by asserting that one is a canonical remainder. No multiplicative-order or RSA theorem is claimed. The published atlas and Alpha membership are unchanged.

Exact theorem in conservative defined notation

∀ m. ∀ i. ∀ v. UnitProductFactor(m,i,v) → Coprime(v,m)

Every linked abbreviation expands hygienically to the identical original native formula.

Definition DAG

UnitProductFactor(m,i,v) · 1Coprime(a,b) · 1

Actual proof prerequisites

coprime_one_left · checked external prerequisite
Original expanded first-order statement 
forall m i v. ((((forall eut_divisor_eu_factor_coprime_coprime. (exists eut_left_eu_factor_coprime_coprime. (i) = eut_divisor_eu_factor_coprime_coprime * eut_left_eu_factor_coprime_coprime) -> (exists eut_right_eu_factor_coprime_coprime. (m) = eut_divisor_eu_factor_coprime_coprime * eut_right_eu_factor_coprime_coprime) -> eut_divisor_eu_factor_coprime_coprime = 1) /\ (v)=(i)) \/ (~(forall eut_divisor_eu_factor_coprime_coprime. (exists eut_left_eu_factor_coprime_coprime. (i) = eut_divisor_eu_factor_coprime_coprime * eut_left_eu_factor_coprime_coprime) -> (exists eut_right_eu_factor_coprime_coprime. (m) = eut_divisor_eu_factor_coprime_coprime * eut_right_eu_factor_coprime_coprime) -> eut_divisor_eu_factor_coprime_coprime = 1) /\ (v)=1))) -> (forall eut_divisor_eu_factor_result_coprime. (exists eut_left_eu_factor_result_coprime. (v) = eut_divisor_eu_factor_result_coprime * eut_left_eu_factor_result_coprime) -> (exists eut_right_eu_factor_result_coprime. (m) = eut_divisor_eu_factor_result_coprime * eut_right_eu_factor_result_coprime) -> eut_divisor_eu_factor_result_coprime = 1)

Complete tactic proof in conservative notation

All 12 original proof lines are preserved. Only local proposition formulas are abbreviated; every abbreviation has an exact binder-safe expansion check. The linked exact edition contains the unchanged replay script.

Read the argument

Proof checkpoints

12 script commands · 7 reading checkpoints · 0 local claims

This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.

Definition notation is shown below. Open the paired exact edition for the original native formulas. Source pairing is not a new equivalence certificate.

01Fix variables and assumptionsL1–4

Work with arbitrary variables or the premises of the current implication.

  1. L1
    intro m
  2. L2
    intro i
  3. L3
    intro v
  4. L4
    intro hf
02Separate the logical casesL5–6

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L5
    cases hf
  2. L6
    cases hf_left
03Calculate and transport equalitiesL7–7

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L7
    rewrite hf_left_right
04Use earlier factsL8–8

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L8
    exact hf_left_left
05Separate the logical casesL9–9

Follow the explicit conjunction, disjunction, witness, or contradiction step recorded below.

  1. L9
    cases hf_right
06Calculate and transport equalitiesL10–10

Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.

  1. L10
    rewrite hf_right_right
07Use earlier factsL11–12

Instantiate or apply named facts and discharge the corresponding proof obligations.

  1. L11
    specialize coprime_one_left (m)
  2. L12
    apply coprime_one_left

Library-wide reading audit

Original defined command ledger · 12 lines
  1. 0001intro m
  2. 0002intro i
  3. 0003intro v
  4. 0004intro hf
  5. 0005cases hf
  6. 0006cases hf_left
  7. 0007rewrite hf_left_right
  8. 0008exact hf_left_left
  9. 0009cases hf_right
  10. 0010rewrite hf_right_right
  11. 0011specialize coprime_one_left (m)
  12. 0012apply coprime_one_left